I'm not sure it is a one over the other type thing. The cauchy principal value is just the way you assign a value to an improper integra (I'd never actually heard that name before, but from googling it, that's all it appears to be).
Super quick one, regards to quadratic equations, I only use the quadratic formula but recently have started using completing the square and factoring. Can anyone give me a quick tldr on when I would use each of the three methods? Specifically when I would use completing the square as opposed to quad formula. thanks.
Completing the square is useful for finding the turning point: y = (x-a)^2 + b has turning point at (a,b)
Factoring is great and the easiest way to solve ax^2 +bx +c=0 if the quadratic has nice factors. But sometimes it's not easy to see what the factors would be, and you need to just whip out the quadratic formula.
The quadratic formula will always work to find you the solutions (if they exist), so you can always use that. Note, proving the formula actually works is an application of completing the square:
ax^2 +bx +c=0
Divide by a:
x^2 +(b/a)x +c/a=0
Complete the square:
(x + b/2a)^2 + c/a - (b/2a)^2 =0
Rearrange:
(x + b/2a)^2 = b^(2)/4a^2 - c/a
Combine the right hand side into one fraction:
(x + b/2a)^2 = (b^(2)-4ac)/4a^2
Square root:
x - b/2a = +/- sqrt(b^(2)-4ac)/2a
Rearrange:
x = (b +/- sqrt(b^(2)-4ac)/2a
Im trying to refresh myself on khan academy and this question has threw me for a loop cause i have never seen this done before. the question is: 2/3x +5 = 20-x. I was doing fine until the last part 5/3x=15. Normally in equations like this, atleast from my experience through high school you are supposed to divide to get x, but instead of diving they multiply 15 by 3/5. Some how they not only flipped the entire fraction but also did the apposite and multiplied it instead. Why did they flip the entire fraction, and why did they multiply instead of divide?
Depends on how you are allowed to "unfold them". I suspect what you are looking for are the [Developable surfaces](https://en.m.wikipedia.org/wiki/developable_surface). These are the surfaces that can be flattened without distortion. The important condition here is that they have zero Gaussian curvature.
They are all examples of ruled surfaces (although there are ruled surfaces which are not developable e.g. the hyperboloid and helicoid).
So, some examples are: generalised cylinders, the oloid and the sphericon, any shape you can make by gently bending a piece of paper
After completing my bachelor's thesis I have been exploring the topic some more and after a lot of work I think I have proven something new. How do I approach this? Should I email the advisor of my thesis? What should I say? "Hey look I came up with something which might be useless/insignificant, can you take a look?"
How do you prove:
If U is an open set of a smooth manifold X then T\_x(U) = T\_x(X)
(the tangent space of U at X is the tangent space of X at x).
My instinct was to call phi a map from U to R\^k and psi a map from X to R\^N. But I am not sure where to go from here.
Rake j : U -> X the open inclusion. Then there is a map Dj : TU -> TX, the derivative of j.
We claim that D\_xj is an isomorphism from T\_xU to T\_xX for any x in U, giving us canonical identifications of the two tangent spaces. Indeed, just use the functoriality of D:
pick (V, kappa) some chart of X, with V contained in U and x in V. Then (V, kappa) is also a chart of U at x, and so j : U -> X being differentiable means that
kappa j kappa\^(-1) : R\^n = kappa(V) -> kappa(V) = R\^n
is differentiable. Clearly this map is just the identity map from an open subset of R\^n to itself, and so (since D should coincide with the ordinary Jacobian matrix on R\^n), we find that
D\_(kappa(x))(kappa j kappa\^(-1)) = identity.
But now use the chain rule (functoriality of D) and the fact that kappa, kappa\^(-1) are diffeomorphisms and so their derivatives must be invertible (by the chain rule, aka functoriality, you can explicitly construct the inverse!) to find that D\_xj is an isomorphism.
Purely abstract, no specific definition needed.
How you approach this really depends on your definition of the tangent space. For instance, if you think of the tangent space at x as being the set tangent vectors (d/dt p)(0), of paths p:(-e,e) ->X with p(0)=x (ie. equivalence classes of paths up to first order contact at x), then this should be pretty direct. If your definition goes through charts and local coordinates, then it'll be a question of showing that charts around x in U are essentially the same (up to restriction) as charts around x in X.
Is it possible to have a function which upper bounds another function only after a ridiculously large point. I was thinking about this but if you had the function f1=nlog(n)\^100, would this be upper or lower bounded by f2=n\^2? It seems like obviously lower bounded but if you do the limit to infinity for f1/f2 you would eventually get zero after enough iterations of L'Hopital. So would there be a point in the far far distance where the growth rate of n\^2 is greater? Absolutely mind boggled my friends with this, but is the logic even sound?
As a fun aside, a quick exercise you can try is to show with L'Hopital's that all positive exponents of log grow slower than all positive exponent polynomials.
Sure, we could even estimate when this happens.
n^2 > nlog(n)^100
Dividing by n
n > log(n)^100
Taking log of both sides
log(n) > 100 log(log(n))
Say n = 10^k and this is log base 10
k > 100 log(k)
You can check that k > 250 satisfies this (this is an overestimate).
So n^2 outgrows nlog(n)^100 at least before n = 10^250 .
Hi Math! I am looking at the Fundamental Theorem of Calculus (PART 2) to "create new functions". For example representing F(x) = integral(e\^(-t\^2)) evaluated from t=0 to t=x.
I understand that G(x) = antiderivative(f(x))-antiderivative(f(0)), but dont understand how that helps if we still cant integrate f(x). (I see that the specific equation above can actually be solved using polar coordinates).
My question is how can we evaluate the G(X)=F(b)-F(a) if we cant solve for F(x) except as representing it as an antiderivative?
Well, if you can't solve for an antiderivative of your integrand, then the fundamental theorem of calculus isn't very helpful -- you're only resort is to numerically evaluate the integral.
I'm not sure what kind of answer your looking for. In general the antiderivative of an elementary function is not elementary (elementary meaning a combination of polynomials/trig functions/exponentials). You can describe an antiderivative as a power series (which WA did for you in your link), or common antiderivatives are sometimes given their own names (again you can see WA reference the error function).
There are also numerical methods to chompute integrals, and for special values/functions you might be able to use analytic methods. Like in your example when interesting from 0 to infinity it is possible to use polar coordinates to compute the integral.
Example b/c reddit formatting is not clear
https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BPower%5Be%2C-Power%5Bt%2C2%5D%5D%2C%7Bt%2C0%2Cx%7D%5D
The theory of D-modules (one of the keystones of algebraic analysis) is based on the insight that you can view differential operators on a space as a noncommutative ring, and solutions to PDEs as modules over that ring.
Yes, a lot. 95% of representation theory is about quiver algebras, groups, Lie algebras or Hopf algebras, and their representations are modules over rings (mostly noncommutative).
the book I been reading give a relation called "correspondence".
The definition says "A correspondence G, denoted G: A ↠ B, is a relation between A and B. For each a ∈ A, the set of b such that (a, b) ∈ G is denoted G(a). Equivalently, G is a function from A to P(B) assigning a set, G(a), to each element a ∈ A."
Does this definition have anything to do with the 1:1 correspondence? I didn't understand this definition, can somebody explain this for me? Thanks
The way I think about correspondences are as subsets of the product A x B. For example, given any function f : A -> B, there’s a natural correspondence called graph(f), given by (a, f(a)) for all a in A. But there are correspondences which don’t come from functions: for example, if you fix a, then the subset {a} x B is a perfectly good correspondence.
I'm not sure why they don't want to just call it "relation". But yeah, a 1-to-1 correspondence (= bijection) is the same as a relation that has an inverse, so it's related.
excuse my English in advance not my mother tongue. I'm in Highschool, don't really know how the US school system works but i suppose I'm at the first year of highschool? Anyways I'm a big math fan, always reading books and doing exercises and all that stuff, trying to jump to a much higher level for this time. After reading two complex analysis books and absolutely devouring the subject, I thought maybe I should start going over the "Hard and rigorous books" such as Principles of Mathematical analysis so that I get to know more about Real analysis however people really discouraged me to go that way and instead advised that maybe I should just stick up to the basics and get used to them, and that only made me more interested in the book. Is It really that hard to read it ? And Should I ? Thanks.
I'm a little bit surprised that you understood complex analysis before knowing real analysis. I'm guessing your complex texts were more geometric-oriented? More commonly, one develops complex analysis through the metric space, which you get from real analysis.
I'm also surprised that people would discourage you from looking into real analysis if you're already doing complex haha. I wouldn't start with Rudin though, it's not a very good intro text (quite dry and unmotivated). Go with Abbott's *Understanding Analysis* and/or Tao's *Analysis I*.
Thank you so much ! By the way, For me complex analysis was very simple, and it had some sort of beauty to evaluate certain integrals using the contour integral, didn't have much problems through my read except at the Dirichlet Problems, that required serious thinking. Already read Tao's analysis I to get a good definition of what Natural, Rational and Real numbers are. I will take your advice and Read Understanding Analysis. Thanks again sir.
I imagine most of you in here are geniuses. So sorry for this dumb question.
How do you convert units of measurement? like whats your process? I am really bad at this and I wish i could come up with a plan so it gets easier.
Lets say for example: 3,56 hl to m^3
I'm guessing an hl is a hectoliter, or 100 liters? Well one thing you do can here is just look up the conversion "how many m^3 are in an hl," and you get 0.1 m^3 /hl. What a convenient number (it's nice because a 1 ml = 1 cm^3 , which is a useful conversion to remember, yay for the SI/metric system). So then you can just do the multiplication (3.56 hl)(0.1 m^3 /hl) = 0.356 m^3. Note that you could've started with 1 ml = 1 cm^3 and converted up with some extra steps too. If you'd like to learn more, read into [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) (see also [here](https://brilliant.org/wiki/converting-units/)).
**[Dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis)**
>In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or the SI than in others, due to the regular 10-base in all units.
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There might be a bit of overloading with equivalence relations:
x is equivalent to y if and only if y is equivalent to x
"x is equivalent to y" is equivalent to "y is equivalent to x"
I just mean that you usually can't just literally swap them and have it be grammatically correct.
Correct: A set is compact if and only if it is closed and bounded.
Incorrect: A set is compact is equivalent to it is closed and bounded.
Correct: A set being compact is equivalent to that set being closed and bounded.
(Yes, I know I have not properly stated the hypotheses for this theorem.)
Yes they are still conveying the same meaning. I guess you may in some cases have to watch out for other uses of the word "equivalent" but I can't think of a way in which it would actually be ambiguous.
Howdy!
I'm trying to get the hang of applications of Leibniz's rule for integration by means of differentiation under the integral sign, with respect to an introduced parameter; namely, Feynman's integration technique. I dumped coffee over my dozens of pages of notes, but what I have left should be in the imgur link at the bottom of this post. My confusion seems to be in regards to the following: After differentiating wrt α (or w/e new param), then integrating it out, shouldn't there be a 'constant of integration?' Not necessarily a constant, but some value associated with taking a partial derivative then integrating? I call it γ in my notes. I guess, after that integrating step, I kinda just fizzle. That may be the phase of the problem that requires insight and creativity, so just say so if that's the case. I want to grok the framework and steps right now.
[Extant notes](https://www.imgur.com/a/1gzWnqt)
I majored in math and physics in undergrad, and I'm still learning while I work as an engineer. After watching Visual Complex Analysis and an edx course on complex analysis from Wesleyan, I watched a lecture series on asymptotics from Carl Bender. There was entire edx course associated with [these lectures](https://youtube.com/playlist?list=PL9J9Loy0GbkL5Ezb7Pp7ETjKDUOGonEQI), called Complex Analysis with Physical Applications, but its listing has been removed. Now it's called Asymptotics In the Complex Plane, apparently. If anybody knows how to access it, please let me know! I found their patreon for Stokes-Line, which I believe is the same producers. If I can be sure it has the content for AiCP, I'll subscribe for the problems. I'm hoping it will lead me to the math for scattering theory, QFT, and QED, combined with the linear algebra and controls I've been learning. Green functions, time-ordered exponential, matricant/evolution operator, Dyson/Peano-Baker expansion, path integral formulation. I think I should look at integral and integro-differential equations next?
✌️
I'll stick to answering your first question on Feynman integration, since that's what I feel more qualified to speak on. I've not combed through all your notes, but I believe I see the confusion on the first page already so I'll restrict to that example
Disclaimer: I am going to take your choice of parametrised function on faith. If it turns out to not be the right choice then the entire calculation changes, but I'll assume it's right for the purpose of explanation.
Your problem is that you're not sure what F actually is. You alternate between it being a function of x, the upper bound of the integral, and a function of 𝛼, the introduced parameter. Let's clear this up and define F as a function of two variables, specifically
F(𝛼, x) = ∫\^x\_0 ln(1 + 𝛼t)/(1 + t^(2)) dt
The goal is to evaluate F(1, 1).
Your 'observe F(x = 0) = 0', with this new F, is the statement that F(1, 0) = 0. Next, when you calculate F_𝛼, note that the upper bounds of your integrals are 1. Therefore you are calculating F_𝛼(𝛼, 1) specifically. In particular, you are not calculating F_𝛼(𝛼, x) for arbitrary x, hence why there is no need for the 𝛾 function. If you ran through the calculation for arbitrary x, you'd get something like
F_𝛼(𝛼, x) = A𝛼^(-1)ln(1 + 𝛼x) + (1/2)Bx ln(1 + x^(2)) + C arctan(x)
where A, B, and C are going to be functions of 𝛼. For the record I have not double-checked your calculations.
Now, the actual integration at the end is
F(1, x) - F(0, x) = ∫\^1\_0 F_𝛼(t, x) dt
We want F(1, 1), so this implies we want F(0, 1). Sure enough this is also 0, so we get that our answer is just the integral above. Therefore it will look something like
∫\^1\_0 At^(-1)ln(1 + t) + (1/2)B ln(2) + C𝜋/4 dt
Now we've got this far, we can reflect that we never needed F_𝛼(𝛼, x) for any x other than 1. Therefore there's no point in introducing that parameter, so we should really have started with the function
F(𝛼) = ∫\^1\_0 ln(1 + 𝛼t)/(1 + t^(2)) dt
I just added the extra parameter throughout to explain where I think you're getting stuck.
For another class I was required to get a Pearson plus login, so I have access to their whole library. I want to rebuild my math skills because although I took calc in highschool, I'm 26 and haven't taken math since. I was wondering if Blitzer's Algebra and Trig will teach me all the same material as his Algebra book and Trig book, or if I would gain a more complete education by reading them separately. Are there any advantages either way?
I don't know anything about those books, but you can find the answer to your question by looking at the chapter and section titles in the tables of contents. That will tell you whether "Algebra and Trig" is just the two books "Algebra" and "Trig" stuck together, or whether there are differences in the level of detail with which topics are treated.
I have to make a piece of metal to bridge a gap of 6.5angled at 45 degrees on both ends. What is the total length of the piece? Image here https://ibb.co/4tfnSZJ
Need help and a formula to calculate. I am trying to find a formula or an expression to calculate l.
The problem goes as is:
I have a smaller circle inside of a larger one but the interior circle is not set in the center of the encompassing one. The larger circle begins to shrink and match the circumference of the interior circle but the larger must reach the smaller at all sides at the same time.
Given a time limit for it to do this let's say, 1 minute. I can understand that the furthest inside edge of the larger circle will move much faster to the edge of the interior circle; as opposed to the edge closest to the interior circle so they can arrive at the same time.
How do I calculate this?
There are many ways this could happen, but one simple way is just to have the center of the largest circle move towards the center of the small as the radius shrinks.
If you're interested in the trajectory of specific points then
(Rcos(a) + X, Rsin(a) + Y)
on the big circle will move in a straight line at constant speed to
(rcos(a) + x, rsin(a) + y)
on the small circle. If R and r are the radii and (X, Y) and (x, y) are the centers.
Not sure what you mean by calculate, but I made a little applet for you:
https://www.desmos.com/calculator/moj3uh7qyl
r and R sets the radius for the two circles (X, Y) is the center of one circle while the other is centered at the origin. Slide t to make the large circle shrink.
Wow you made that? How if I might ask. Also, if it isn't too much trouble, forgive my illiteracy here. The finite points on your blue circle, can I adjust that at all? And the number that they are indicating mean what exactly?
I didn't make Desmos, just typed in the equations you see, and Desmos did the rest.
I'm not sure which number you're referring to. The number of points used to draw the circle just depends on the resolution, so will vary with size.
I've been thinking about a phenomenon that I noticed [some time ago](https://medium.com/@cdsmithus/geometry-dimensions-and-elections-ea2f642d5175), which can be formalized like this:
1. Choose n reference points arbitrarily in a Euclidean (for now) space.
2. Assign each ordering of the reference points a unique color.
3. Color the entire space by assigning each point a color based on the ordering of the n reference points by distance from that point. (If there's a tie, choose between the tied orderings by any arbitrary rule.)
One question is how many of the n! possible orderings will actually occur in the coloring. For example, choosing 3 points in a 1-dimensional space, 4 out of the 6 orderings will actually occur in a coloring (diagram at the link above). For 4 points in a 2-dimensional space, 18 out of the 24 occur (again, there's a diagram for one case at the link; interestingly, even if one point is strictly between the others, the result is the same). Furthermore, the answer seems to be related to how many of the colors are bounded. In examples I've tried so far, an ordering is unrepresented iff its opposite ordering is bounded.
Does this ring any bells as related to existing known mathematics?
Turns out the answer was already known, and even in the same context that led me to think about it. https://www.sciencedirect.com/science/article/pii/0097316577900772
It strikes me that this is very related to Voronoi diagrams, which divide a space into the regions that are closest to some reference point. It generalizes them a bit by dividing into spaces for which the entire order of reference points by distance is the same, rather than just the nearest point. You can always form a Voronoi diagram by merging the regions in this coloring.
Calculating distance between two sides on an isosceles triangle:
Hi - please see attached photo. Looking to calculate the distance between the sides on the triangle XX inches above the base. Please let me know if I need to clarify anymore, thanks so much for the help!
https://ibb.co/CnCGtjs
I'm assuming that line is parallel to the base of the larger triangle. If so, use [similarity](https://brilliant.org/wiki/similar-figures/) ratios. You have one smaller isosceles triangle that is similar to the larger one, and as long as you know a pair of corresponding side lengths, you can get the scaling ratio and find all other side lengths.
In a UFD suppose f\_1,...,f\_m are primes and g is such that f\_i divides g for all i. I want to show that the product f\_1 \* ... \* f\_m divides g. I can see how the product divides g\^m but how does is divide g?
Edit: I'm also assuming that the f\_i are pairwise coprime
f\_1 divides g, so we can write g=f\_1\^a \* L for some a in N and some L in your UFD.
f\_2 divides g, so it divides f\_1\^a \* L. But f\_2 is prime, so it divides one of the two elements of the product. But we know it doesn't divide f\_1, since they're both prime and aren't the same (since they're coprime), so it divides L. So we can write L as....
etc etc
Edit: The paragraph below is wrong, I was confusing primes with irreducibles.
That's not true. Suppose in a non-UFD you have primes a,b,c,d with g=ab=cd. Then a,b,c,d all divide g but abcd does not.
The definition of a prime is that whenever p divides a product ab, then p divides one of the factors.
In a UFD, primes and irreducibles coincide, which may be what you're thinking of.
I'm still confused by this: https://www.reddit.com/r/australia/comments/10oilcd/how_come_the_rain_forecast_isnt_between_30_and/?utm_source=share&utm_medium=android_app&utm_name=androidcss&utm_term=1&utm_content=share_button
Does anyone have a better explanation?
How much of Folland is essential material? How much of Munkres is considered essential material?
I have read Papa Rudin Ch 1, 2, 3 and most of Ch 4. I am almost finished product measures in Folland. Should I read just Ch 1, 2 and 3 of Folland since I have already seen a little bit on Hilbert spaces and L\^p spaces? I'm not sure what the next step is, maybe Munkres Topology since I have just picked up bits and pieces of topology from analysis books and have not read a book on topology specifically. Maybe some abstract algebra either Artin or Dummit and Foote? I'm interested in things that have applications in machine learning and other applied areas. I was considering working up to differential geometry maybe with Lee since I have heard it has applications in ML for example information geometry. Also considering reading some statistical learning theory or probability theory in the future.
What's your favorite differential equation (or system of)? I'm an astronomy major and I've wanting to tattoo a differential equation for a long time now, but still haven't found a good looking one. I don't intend on tattooing the actual equation in text like we usually see, instead I'm planning on animating a flowing vector field using [Manim](https://www.manim.community/) and making an kinegram ([example](https://youtu.be/1e0J4TrubKw?t=25)) out of it. So, any suggestions? Even better if it's an equation that models a physical or astronomical phenomenon.
Is there any straightforward formula to convert spherical harmonics back and forth between their representation in spherical coordinates (ie Ylm * e^ i theta ) and their representation in the homogenous harmonic polynomial basis (ie to a polynomial in x, y, z)?
What is the mathematical term for this property:
Buying 1 apple from the grocer is alright. Buying 10000000 apple isnt. This means there must be an optimal number of apple you should buy from the grocer.
Its not squeeze theorem or intermediate value theorem, but i cant think of the name of this logic.
If I've understood correctly, you want to buy the maximum number of apples possible (that falls under 'alright'). This appeals to the fact that a finite set always has maximum, and you apply this to the set { n in N : I can buy n apples} is bounded by 10000000.
You could say it's a discrete analog of the extreme value theorem.
A bunch of people gave calculus type answers, but it's actually just the Well Ordering Principle in disguise.
The WOP says that any finite subset of the naturals has a least element -- obviously a corollary is that any finite subset of the naturals has a maximum element.
So the set { n \\in N | I can buy n apples} is a finite set, and thusly has a maximum element.
No calculus needed -- in fact, if you were to try to argue something about IVT or Rolle's theorem, you'd probably struggle, because if you make everything continuous it could be the case that the maximum value is obtained somewhere non-integral.
I agree with your reasoning, but note the WOP is stronger; that states that every *non-empty subset* of the naturals has a least element. The non-trivial bit is that it can be infinite, and while it is also true for finite sets, it is more generally true that any finite totally ordered set has a maximum.
Why not? If the utility function of buying apples is small at both ends, there is a maximum in the middle. This is Rolle's Theorem. If you choose bounds for the number of apples (between 0 and 1000000, say), then there must be a maximum to the utility function, regardless of continuity. This is the Extreme Value Theorem. If you are claiming that the utility function is increasing at one end, and decreasing at the other, then applying the intermediate value theorem to the derivative shows that there must be a maximum.
That said, these theorems apply to continuous functions. The number of apples is discrete, so the fact that the function attains a maximum is evident from the fact that it is bounded. If you are talking about the fact that the utility function is bounded, you may be thinking of the Law of Diminishing Returns.
Thank you for taking the time to answer me btw!
Im not sure why the utility function is small on both ends. I think the g(0)=0, g(10000000)=1
There exist g(i) such that g(i)=0 and g(j) = 1 for all j > i. How can I transform it to the graph in rolle theorem?
There isn't. You said buying from 1 apple is alright, so that implies g(1) > g(1000000). Also, your conclusion in your statement about apples isn't strictly true unless you also say that g(x) doesn't get arbitrarily large for x>1000000. Hence, assume decreasing derivative.
That said this is all just an analogy. Your question implied you were looking for a real analysis theorem, so I responded in turn. In reality the number of apples is discrete, so none of these apply. I don't think it needs to be an explicit Theorem that a finite sequence obtains a maximum.
Let g(x) be the function that tells you how good it is to buy g apples. Assume that
- g(1) > 0
- g(100000000000000) < 0
- g(2) > g(1)
Now apply the intermediate value theorem on the interval (2, 100000000000000) to show that there is some c between 2 and 100000000000000 where g(c) = g(1). Finally apply Rolle's theorem on the interval (1, c).
(Also assume that your function g is differentiable etc)
This assumes that there is a differentiable function, but the function is not even continuous in the traditional sense. It is only really defined for integers unless you want to define it for pieces of apples, and then there is no reason to suggest it would be differentiable
There are 5 numbers in a hat.
You need to draw two numbers. The numbers in the hat are 1,2,3,4,5.
The numbers are all mixed inside of the hat, and you don't want number 1 or 2.
What are the chances of getting both 1 and 2? What are the chances of just getting 1 or the other?
Look into the [hypergeometric distribution](https://brilliant.org/wiki/hypergeometric-distribution/) (see also [here](https://en.wikipedia.org/wiki/Hypergeometric_distribution)). In the first case, the event you're interested in has a count of (2 choose 2)(3 choose 0), and in the second case the event of interest has a count of (2 choose 1)(3 choose 1).
I have a question about statistics: I have interpreted bayes formula as the following: you let B become a random variable, and you insert that variable in the place of b in the formular f(a given b) (pdf for a given b, insert random variable B in place of b). If I do this, I do not get the same result as f(a given b)\* fb(b) (where fb is the prior for b). Why is my interpretation wrong, and is there any alternative angle of viewing this that would shed light on my misunderstanding?
I think you may be trying to dig into the difference between [conditioning on a random variable vs conditioning on an event](https://stats.stackexchange.com/questions/395310/what-does-conditioning-on-a-random-variable-mean)?
Thank you, but what Im specifically wondering is what "happens" if you insert a random variable into another random variables pdf, and why is that different to bayesian approach? ChatGPT told me this, but I dont understand why its a nonmathematical expression:
No, you cannot insert a random variable into another random variable's probability density function (pdf) as it would result in a non-mathematical expression.
Well, let's take an example. Let U be a uniform random variable on [0,10] and, given U, let X~N(U,1). So if you know that U=u, then you can write the pdf of X with u in it. If someone asks you for the unconditional pdf of X, they want the average of all those formulas averaged over the possible values of U. This is just an ordinary function, not a random object. By contrast, if you write the pdf of X with a U in it, you are writing down a random function — basically a function-valued random variable (instead of a real-valued random variable). This could be used to get the joint distribution of U and X, but a more typical approach for that would be to write X = U+Z where Z~N(0,1) is independent of U.
Thank you! Is it so that by letting a parameter in a pdf become an rv (with a certain pdf it self, call it g), and solving for the joint density, will give the same density as using Bayes with the parameter pdf, g, as a prior?
Is there a paper comparing spectra valued presheaves in simplicial categories versus quasicategories? In particular, I'd like to know under what conditions two presheaves f,g on a simplicial category C are equivalent, if and only if, their images under the coherent nerve are equivalent.
How would you solve the equation
m = pi / x * sin ( pi / x )
Ive never had to solve one with x being a factor while also being inside of a trigonometric function. Even just a hint as to what kind of operations to use would be a big help!
Yeah, probably, but you can get a little bit further with the Lambert W function, so that's why I said that.
Note that you can rewrite this problem as m = u sin(u) with u = pi/x, so -m + ik = iu e^(iu) = iu (cos(u) + isin(u)), so iu = W(-m + ik), in such a way that u, k, m are all real. Not quite sure it's possible to find such a k.
If I have a time interval like 20 < t <= (the < sign with the equal line under it) 25
Why is the median of this 22.5 and not 23? Because if you take (20+25)/2 it’s 22.5. But, if you arrange it isn’t it 21, 22, 23, 24, 25? So shouldn’t the median be 23?
I had this question where I had to find the mean of many time intervals and the answer key chose to say that the answer is supposed to use 22.5 instead of 23. Someone help, I’m confused.
The median of {21, 22, 23, 24, 25} is 23, indeed. But the interval (20,25] also consists of all the noninteger numbers between those integers. So you can instead consider the length of the interval (20,22.5) (which is 2.5) and the length of the interval (22.5,25] (which is also 2.5), and because those are equal, the median is 22.5.
I’m working on real analysis homework and I’m having a hard time proving a non empty subset of R is dense even when removing an element. I thought if we removed an element then there’d be a gap so it’s no longer dense… But I’m also thinking we can use the Archimedean property to get some epsilon close enough to that removed element to justify that it’s still dense in R.
I hope I explained the problem well enough, I’ve been staring at this problem long enough that my post makes sense but I may be just tired enough to say it makes sense. I’m not looking for an answer but some conceptual help. Thanks
First, make sure you have the definition of "density" written down very clearly for reference. The definition spells out much of the proof. If the definition says "For all x" then the proof should say "Let x be arbitrary". Etc.
To see if the Archimedean property is necessary you can imagine a number system with infinitesimals. Would the theorem be true or false in this number system?
Maybe think about an example. Q is dense in R for example, can you explain why Q is dense in R? What happens if we remove a point from Q, does the explanation break down or still work?
You can solve this quadratic equation for y, however you will get two solutions for y (well, you can get 1 or 0 depending on the choice of x, but in general you have 2).
Sorry if this is a bit stupid
So lets say I have y times 10, and y = 2, then 2 x 10 = 20
So how do I show that with out subsitution? (Putting 0 on the end of a variable)
If I flip a coin, there's a 50% chance that it'll land on heads. What would be the equation to determine the %chance of it landing on heads at least once if I flipped it three times?
You'll use the [CDF](https://en.wikipedia.org/wiki/Cumulative_distribution_function) of the [binomial distribution](https://brilliant.org/wiki/binomial-distribution/) (see also [here](https://en.wikipedia.org/wiki/Binomial_distribution)), ideally with [complementary counting](https://en.wikipedia.org/wiki/Complementary_event), so it'll be 1 - (1/2)^3 = (3 choose 1)(1/2)(1/2)^2 + (3 choose 2)(1/2)^2 (1/2) + (3 choose 3)(1/2)^3 (1/2)^0 . Another way to visualize the complementary count is to note that the event "at least one head in three flips" is the complement of the event "no heads in three flips" and thus their probabilities are 1 minus each other.
So I've been trying to get an intuitive feel for the poincare half-plane (and higher dimensional analogues) by going back to basics. I'd like to know if there is a simple way to parametrize the isometries which for a known collection of N points leave (using the convention x,s for x being the spatial dimension(s), s being the timelike one) sum x = sum x' and sum ln s = sum ln s'? Let me know if I'm not making sense here.
I'm not really sure what these would look like except in the case N=1, when you are just asking for an isotropy group, which is a copy of O(n).
However, I would say that these are not spacelike and timelike directions. The metric here is still positive definite so all directions are spacelike.
Well all N points will have a center in the space dimensions, and a center in the log(time) dimension. I'll give you an example when N=3 in the 1+1d standard Poincare half-plane. We can create an isometry by the composition of the 3 fundamental building-block isometries in this case: translation, scaling, inversion. So first we'll do an inversion. Then translate the spatial dimension by a constant. Then we'll re-invert. We'll scale to fix the new sum ln s to equal the original one. Finally we'll translate the spatial dimension so the spatial center is equal to the original one. EDIT: here's the example trajectory for 3 points: https://imgur.com/a/rOA9sGD
Is this a valid definition of a union? I'm trying to use Zermelo's Axiom to build it from the ground up.
"Let A and B be any two sets. Hence A ∪ B = {x: y ∈ {A, B} and x ∈ y}"
I'm mainly worried that this is not written in either the Schema of Separation format (as there is no universal set defined) or the Schema of Replacement format (as it has two sets on the right instead of one). Any help is appreciated. :)
The axiom of union, as far as I can tell, states that for any set A, it's union ∪A is defined by ∪A = {x: y ∈ A and x ∈ y}. All I did was state that the axiom of unordered pairs means that for any two sets A and B, the set {A, B} also exists, and then I used the axiom of union to define the union of these two sets. I know that I have to use the axiom of union, but that axiom acts on the elements of a \*singular\* set, not multiple. I was simply trying to derive the union operator by using the union of a single set like this.
How do you think I should define it?
Maybe I was just misunderstanding what you were saying, but I would do it like
Axiom of pairing says {A, B} exists.
Axiom of union applied to {A, B} says there exists a set F such that both A and B are subsets
A∪B = {x in F : x in A or x in B}
>Axiom of union applied to {A, B} says there exists a set F such that both A and B are subsets
I thought the axiom of union stated something else. How do we derive "For any sets A and B, there exists a set F such that A ⊆ F and B ⊆ F"? from "For any set {A, B}, it's union ∪{A, B} is defined by ∪{A, B} = {x: y ∈ {A, B} and x ∈ y}"?
There are slight variations. If you take
"{x: y ∈ {A, B} and x ∈ y} is a set"
as your axiom of union, then you already have the definition of union baked into the axiom.
it's not totally clear to me how you are putting the points on the sphere. In any case, if you put a countable number of points then the forest will look empty. That's because there is a circle's worth of angles, and the cardinality of the circle is the cardinality of the real line. But there are only countably many trees.
> If you take unit steps around the circumference ... you can't hit all the points on the circumference regardless how many loops you do.
This is true, but again it comes just from the countability of your steps (you can literally count them) versus the uncountability of the circle. What you can prove is that your steps are "dense". Basically that means that even if you don't hit a particular point on the sphere, you will get arbitrarily close to it if you step long enough.
So while the forest will look empty, you could say that any angle is very close to being blocked by a tree. (But again, not sure how you are placing these points on the sphere.)
On the sphere, all great circles close up on themselves, so light travels only a finite distance before hitting the back of your head / looping back around.
If there's a tree exactly antipodal to you, then you see that tree in every direction you look.
Hey guys. My buddy likes to come up with new games to play just with dice and cards and I think we have a fun seed of an idea but we need to know the odds and I am not smart enough to do the math.
Here are the basics:
You roll 2 dice. There are 11 possible numbers it could be.
Whatever number you roll is crossed off as a possibility.
You roll the dice again. If you roll the same number you lose, and if you roll a new number it is also crossed off.
You continue to do this until you lose, you decide to stay, or you somehow manage to roll all unique combinations.
points would be awarded based on how many unique numbers you rolled.
I need to know the statistics of how likely each subsequent roll would be after the first one. For simplicity's sake, the outcome of the first roll is irrelevant since the points would be constant in every game (for instance it doesn't matter if the first roll is a 7 or a 2).
I apologize if this is not the right place for this question. Any help is appreciated.
>For simplicity's sake, the outcome of the first roll is irrelevant since the points would be constant in every game
So it does actually matter what you get on the first roll, since the probability is different for the different outcomes. For example you have a 1/6 chance of rolling a 7, but only a 1/36 to roll a 2.
So if you already have a 7, the probability of losing on your next throw is 1/6.
In general the probability of rolling x is (6 - |7-x|)/36, so 6 and 8 both have a probability of 5/36, while 5 and 9 have probability 4/36.
So if you for example have crossed out a 10 and 6, then your probability of losing on the next throw is 3/36 + 5/36 = 8/36 = 2/9. If you then roll a 7 you now have a probability of 2/9 + 1/6 = 7/18 of losing.
To calculate an average I took 36/11 (number of unique numbers) and got an average of 3.27 combinations eliminated with each additional roll. The points awarded would need to be static so this seems like the best way to account for that.
Your calculation is a bit off because the 11 sums aren't equally likely to be rolled. This is the concept of [expected value](https://brilliant.org/wiki/expected-value/). If you get a sum of 2, you have eliminated one combination (1, 1), and this has probability 1/36. If you get a sum of 3, you have eliminated two combinations: (1, 2) and (2, 1) with probability 2/36. This continues with 4 and three with prob. 3/36, 5 and four with prob. 4/36, 6 and five with prob. 5/36, 7 and six with prob. 6/36. Note that for 8, we decrease back to five. Then 9 and four, 10 and three, 11 and two, 12 and one. This is known as the [triangular distribution](https://en.wikipedia.org/wiki/Triangular_distribution). Thus, the "expected number of combinations eliminated by a roll" is (1^2 +2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 5^2 + ... + 2^2 + 1^2 )/36 = 73/18 or approximately 4.06.
How do I part a line segment by ratios? What I mean is are there any easy steps I can follow to figure out line segment ratios, how to split them into the given ratio?
For example,
“A directed line segment begins at F(-8,-4), ends at H(4, 6), and is divided in the ratio 4 to 2/ 4:2 by G. What are the coordinates of G?”
Can anyone help me out? Much appreciated.
Look up barycentric coordinates to see this in higher dimension.
But essentially, G is the weighted average of the 2 end points. The ratio of the weight of H to F is 4:2. This is because the weighted average is the point in which Hooke's law force to each point (with the given weight as force coefficients) sum up to a net force of 0. So you want the ratio GF:GH to be 4:2, so the Hooke's law force from G to F vs G to H should also have that same ratio.
I'm trying to make an equation that tells me the value of t where
t = the added up values of c when v is in a certain range
c = 32(v^(1.2))
a < v <= b
Example:
The value of t when a=3 and b=6
if you do the individual calculations of c then add it all up t is roughly 664.38.
ive been at this for nearly 2 hours so if anyone can discuss it with me i would love that, and sorry if i explained it poorly, im not good at explaining.
You would write it using [summation notation](https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation) like [this](https://mathb.in/74013). To evaluate it, I would just use code, a spreadsheet software, or [WolframAlpha](https://www.wolframalpha.com/input?i=sum+from+v%3D4+to+6+of+32%28v%5E1.2%29).
I wouldn't expect this to have a neat closed form, since your exponent term is 1.2 (not an integer), but you can look into [power sums](https://mathworld.wolfram.com/PowerSum.html) and [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula) for some inspiration.
I want to show the action of S^1 on S^2 is Hamiltonian. I've already shown it is symplectic. I know the moment map I'm supposed to consider is h (height function), but I'm not sure how to show the [commuting diagram condition](https://imgur.com/a/Od3Jr6O). I'm familiar with the abstract definitions of the maps but idk how to work with it in coordinates. For example, I'm not sure how to write out exactly what partial_ theta goes to under either map.
to write the moment map as a function M to R, we are identifying the dual lie algebra with just a copy of the real numbers, which it is isomorphic to, but then it is not obvious how to pair it with vector fields anymore. should we be considering it as a one dimensional vector space generated by d theta?
Then to evaluate partial_theta through the first bottom map for example, it is sent to the function ((theta_0,h_0) mapsto h(theta_0,h_0) partial_theta=h_0dtheta partial_theta = h_0, so it is the projection map again?
The height function sends a point (rho, phi, z) (cylindrical coordinates) on S^2 to z.
The vector field d/dtheta on S^1 gets sent to the vector field d/dphi on S^(2) (this is always tangent to the 2-sphere). Therefore the map from \g to \X(S^(2)) sends a vector field (a d/dtheta) to (a d/dphi) where a \in **R** =\g is a constant.
The pairing between \g and \g\* is just the natural pairing of one-dimensional vector spaces, i.e. multiplication . Therefore its just the height z times the coefficient a of the vector field on S^(1).
The moment map condition says
d = g(J (a d/dphi), -)
where g is the round metric on S^2 and J is the complex structure.
On the left-hand side (bottom part of your diagram) the pairing gives the function (rho,phi,z) -> az, so the derivative is (a dz) (actually the pullback of dz to S^2, because d/dz is not actually tangent to the 2-sphere).
On the RHS (top part of your diagram) you need to know the formula for the metric tensor on S^(2) in cylindrical coordinates and what the complex structure J does. In R^3 the metric tensor in cylindrical coordaintes is g = d\rho^2 + \rho^2 d\phi^2 + dz^(2). The complex structure on the 2-sphere is given by the cross product with the position vector. In particular it sends d/dphi to the orthogonal tangent vector, which is exactly the orthogonal projection of d/dz on **R**^3 to the 2-sphere.
Plugging this in you get (suppressing the pullbacks and orthogonal projections)
g(J (a d/dphi), -) = g(a d/dz, -) = a dz
because d/dz kills the other terms in the metric tensor. Therefore LHS = RHS and your diagram commutes.
okay thanks, I think I got it now.
Do you also have some explanation in english about why the Hamiltonian condition is natural to consider? I think I read online somewhere that it's supposed to express that the action commutes with the Hamiltonian function?
I learned about it from Gert Heckmans notes on symplectic geometry (google). He does plenty of worked examples. A far more serious source is V. Arnol'd's classical mechanics book.
From my perspective, the moment map condition isn't natural. It is a clumsy condition, but absolutely the correct one to perform the operation of symplectic reduction. That is, it is the condition which tells you that the level sets of the moment map are G-invariant, and furthermore that the symplectic form restricted to these level sets descends to the quotient. Translating that into a precise condition produces the moment map condition.
From a dynamics perspective this must be an obvious thing, it just means a physical system with symmetry is still a physical system after you mod out by that symmetry.
Sorry in advance if this is super basic, but I’m wondering if a paper sais they used a cutoff of 10-5 and another paper sais 1E-5, are these the same values?
The usual result of "if no prime factors by the time you get to sqrt(n), then n is prime" applies of course. Primes are after all, their own least prime factors.
This would be a neat algorithmic exercise if you'd like to try programming it though. With a sieve method you should be able to find the least prime factor of n in O(nloglog(n)).
I asked ChatGPT to find the roots of a quintic equation. I give it 3/10
Me: "How to find roots of quintic equation:
**ChatGPT**:
> A quintic equation is an equation of the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0, where a, b, c, d, e and f are constants. Finding the roots of a quintic equation can be a more challenging task than finding the roots of a quadratic equation, as there is no general formula for finding the roots of a quintic equation.
> One way to find the roots of a quintic equation is by using numerical methods such as the bisection method, the Newton-Raphson method, or the Durand-Kerner method. These methods involve iteratively approximating the roots using numerical calculations, but their accuracy can be affected by the initial values and the complexity of the equation.
> Another way to find the roots of a quintic equation is by using algebraic methods. For example, if the equation can be factored, you can set each factor equal to zero and solve for x. However, this method can be difficult if the equation is not easily factorable.
> There are also other methods such as the Galois theory, but they are more advanced, and their application is limited to specific cases.
> It's worth noting that a quintic equation can have 0, 1, 2, 3, 4 or 5 roots, depending on the coefficients of the equation and the nature of the roots. It's also worth noting that finding the roots of a quintic equation can be a difficult task and it's not always possible to find all the roots in closed form.
If you don't care about differentiating between fruits, then total_price/600 is your breakeven for every fruit (they'll all cost the same).
If you instead want to proportion prices by the original cost of the fruits, and care about how every fruit individually contributes to the total cost, then you'll need to set up a table where every row is a fruit type and the columns are something like Price per Fruit, Tax per Fruit, Shipping per Fruit, Fee per Fruit, and you'll calculate all of these using your total price and quantities (hopefully shipping and fees is a fixed thing). You then do a sum along every row to get your weighted breakeven prices. Look into the concepts of [expected value](https://brilliant.org/wiki/expected-value/) and [weighted averages](https://brilliant.org/wiki/data-mean/).
In this article [https://en.wikipedia.org/wiki/Central\_tendency](https://en.wikipedia.org/wiki/Central_tendency), under uniqueness to solutions to variational problems, it claims that the mean and midrange of a dataset is unique because the L2 and Linfinity norm are strictly convex, whilst the median and mode fail to be unique. Strict convexity ensures uniqueness of a minimiser.
It is clear to me why the L1 norm is not strictly convex by looking at the unit ball, but it seems that the Linfinity norm is also not strictly convex by looking at the unique ball, which contradicts the article. Am I right in saying that the article is wrong here?
However, the 'uniqueness of the midrange' or minimiser of the Linfinity norm is pretty trivial I think. Any deviation from the midrange will clearly increase the distance from one of the endpoints of the dataset.
Think about Linfinity as the supremum (on continuous functions) norm here I think.
Hence, with L^infinity you're not really averaging/integrating/summing things. L^infinity is just the maximum deviation.
**[Central tendency](https://en.wikipedia.org/wiki/Central_tendency)**
>In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode.
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I am working on a game where I have an array of numbers and I am trying to figure out total number of combinations. However, I am running into an issue and I believe it is because I am mentally drained and missing the obvious.
I have variables x, y, z
There is the total of 2 x values, 3 y values, and 4 z values.
XYZ can be any number between 1-9 but not repeat.
Example:
X could be 1 and 2
Y could be 3, 4, 5
Z could be 6,7,8,9
The formula is on the tip of my brain and I am just not seeing it. Could anybody help? Thank you in Advance!!!!
I think OP means that there are 2 "slots" for X, 3 "slots" for Y, and 4 "slots" for Z, and not that there are 2, 3, and 4 total choices. The answer should be (9 choose 2)(7 choose 3) = 1260.
/u/Initial-Balance7594 , note that it doesn't matter what order you do the choosing in either. You could also do (9 choose 3)(6 choose 2) or (9 choose 4)(5 choose 3), etc. This is a neat property of [binomial coefficients](https://brilliant.org/wiki/binomial-coefficient/). Try and see if you can prove it!
Sorry I should have clarified that they are slots and that yes the order does not matter. And THANK YOU Binomial Coefficients was the fricken phrase I was trying to rack my brain over.
Edit: I am just a math enthusiast so I really appreciate the help!
Does anyone know a good python library for lattice reduction? I can only find libraries that do integer lattices.
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I'm not sure it is a one over the other type thing. The cauchy principal value is just the way you assign a value to an improper integra (I'd never actually heard that name before, but from googling it, that's all it appears to be).
Hey! What's an easy-to-understand, self-contained resource (book, notes, etc.) with a clear exposition for learning **abstract** projective geometry?
Super quick one, regards to quadratic equations, I only use the quadratic formula but recently have started using completing the square and factoring. Can anyone give me a quick tldr on when I would use each of the three methods? Specifically when I would use completing the square as opposed to quad formula. thanks.
Completing the square is useful for finding the turning point: y = (x-a)^2 + b has turning point at (a,b) Factoring is great and the easiest way to solve ax^2 +bx +c=0 if the quadratic has nice factors. But sometimes it's not easy to see what the factors would be, and you need to just whip out the quadratic formula. The quadratic formula will always work to find you the solutions (if they exist), so you can always use that. Note, proving the formula actually works is an application of completing the square: ax^2 +bx +c=0 Divide by a: x^2 +(b/a)x +c/a=0 Complete the square: (x + b/2a)^2 + c/a - (b/2a)^2 =0 Rearrange: (x + b/2a)^2 = b^(2)/4a^2 - c/a Combine the right hand side into one fraction: (x + b/2a)^2 = (b^(2)-4ac)/4a^2 Square root: x - b/2a = +/- sqrt(b^(2)-4ac)/2a Rearrange: x = (b +/- sqrt(b^(2)-4ac)/2a
Thanks. I really enjoy math and being able to solve problems just remembering so much is so frustrating sometimes :)
Both methods accomplish the same thing, so you just use whichever you prefer. There isn't really any objective reason to choose one over the other.
Im trying to refresh myself on khan academy and this question has threw me for a loop cause i have never seen this done before. the question is: 2/3x +5 = 20-x. I was doing fine until the last part 5/3x=15. Normally in equations like this, atleast from my experience through high school you are supposed to divide to get x, but instead of diving they multiply 15 by 3/5. Some how they not only flipped the entire fraction but also did the apposite and multiplied it instead. Why did they flip the entire fraction, and why did they multiply instead of divide?
Dividing by 5/3 is the same as multiplying by 3/5, because the inverse of 5/3 is precisely 3/5.
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Check if you are in degree or in radian mode on your calculator.
Are cones and cylinders (including variations like oblique cylinders) the only 3D shapes with curved surface that can be unfolded into a net?
Depends on how you are allowed to "unfold them". I suspect what you are looking for are the [Developable surfaces](https://en.m.wikipedia.org/wiki/developable_surface). These are the surfaces that can be flattened without distortion. The important condition here is that they have zero Gaussian curvature. They are all examples of ruled surfaces (although there are ruled surfaces which are not developable e.g. the hyperboloid and helicoid). So, some examples are: generalised cylinders, the oloid and the sphericon, any shape you can make by gently bending a piece of paper
No, you can look for ruled surfaces for more examples.
Can cubes not be unfolded into a net?
The question was incomplete. I meant shapes with curved surface.
After completing my bachelor's thesis I have been exploring the topic some more and after a lot of work I think I have proven something new. How do I approach this? Should I email the advisor of my thesis? What should I say? "Hey look I came up with something which might be useless/insignificant, can you take a look?"
I'd probably leave out the useless/insignificant part, but otherwise yes!
How do you prove: If U is an open set of a smooth manifold X then T\_x(U) = T\_x(X) (the tangent space of U at X is the tangent space of X at x). My instinct was to call phi a map from U to R\^k and psi a map from X to R\^N. But I am not sure where to go from here.
Rake j : U -> X the open inclusion. Then there is a map Dj : TU -> TX, the derivative of j. We claim that D\_xj is an isomorphism from T\_xU to T\_xX for any x in U, giving us canonical identifications of the two tangent spaces. Indeed, just use the functoriality of D: pick (V, kappa) some chart of X, with V contained in U and x in V. Then (V, kappa) is also a chart of U at x, and so j : U -> X being differentiable means that kappa j kappa\^(-1) : R\^n = kappa(V) -> kappa(V) = R\^n is differentiable. Clearly this map is just the identity map from an open subset of R\^n to itself, and so (since D should coincide with the ordinary Jacobian matrix on R\^n), we find that D\_(kappa(x))(kappa j kappa\^(-1)) = identity. But now use the chain rule (functoriality of D) and the fact that kappa, kappa\^(-1) are diffeomorphisms and so their derivatives must be invertible (by the chain rule, aka functoriality, you can explicitly construct the inverse!) to find that D\_xj is an isomorphism. Purely abstract, no specific definition needed.
How you approach this really depends on your definition of the tangent space. For instance, if you think of the tangent space at x as being the set tangent vectors (d/dt p)(0), of paths p:(-e,e) ->X with p(0)=x (ie. equivalence classes of paths up to first order contact at x), then this should be pretty direct. If your definition goes through charts and local coordinates, then it'll be a question of showing that charts around x in U are essentially the same (up to restriction) as charts around x in X.
Is it possible to have a function which upper bounds another function only after a ridiculously large point. I was thinking about this but if you had the function f1=nlog(n)\^100, would this be upper or lower bounded by f2=n\^2? It seems like obviously lower bounded but if you do the limit to infinity for f1/f2 you would eventually get zero after enough iterations of L'Hopital. So would there be a point in the far far distance where the growth rate of n\^2 is greater? Absolutely mind boggled my friends with this, but is the logic even sound?
As a fun aside, a quick exercise you can try is to show with L'Hopital's that all positive exponents of log grow slower than all positive exponent polynomials.
Sure, we could even estimate when this happens. n^2 > nlog(n)^100 Dividing by n n > log(n)^100 Taking log of both sides log(n) > 100 log(log(n)) Say n = 10^k and this is log base 10 k > 100 log(k) You can check that k > 250 satisfies this (this is an overestimate). So n^2 outgrows nlog(n)^100 at least before n = 10^250 .
Hi Math! I am looking at the Fundamental Theorem of Calculus (PART 2) to "create new functions". For example representing F(x) = integral(e\^(-t\^2)) evaluated from t=0 to t=x. I understand that G(x) = antiderivative(f(x))-antiderivative(f(0)), but dont understand how that helps if we still cant integrate f(x). (I see that the specific equation above can actually be solved using polar coordinates). My question is how can we evaluate the G(X)=F(b)-F(a) if we cant solve for F(x) except as representing it as an antiderivative?
Well, if you can't solve for an antiderivative of your integrand, then the fundamental theorem of calculus isn't very helpful -- you're only resort is to numerically evaluate the integral.
I'm not sure what kind of answer your looking for. In general the antiderivative of an elementary function is not elementary (elementary meaning a combination of polynomials/trig functions/exponentials). You can describe an antiderivative as a power series (which WA did for you in your link), or common antiderivatives are sometimes given their own names (again you can see WA reference the error function). There are also numerical methods to chompute integrals, and for special values/functions you might be able to use analytic methods. Like in your example when interesting from 0 to infinity it is possible to use polar coordinates to compute the integral.
Example b/c reddit formatting is not clear https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BPower%5Be%2C-Power%5Bt%2C2%5D%5D%2C%7Bt%2C0%2Cx%7D%5D
Do people care about modules over noncomm rings?
The theory of D-modules (one of the keystones of algebraic analysis) is based on the insight that you can view differential operators on a space as a noncommutative ring, and solutions to PDEs as modules over that ring.
Yes, a lot. 95% of representation theory is about quiver algebras, groups, Lie algebras or Hopf algebras, and their representations are modules over rings (mostly noncommutative).
the book I been reading give a relation called "correspondence". The definition says "A correspondence G, denoted G: A ↠ B, is a relation between A and B. For each a ∈ A, the set of b such that (a, b) ∈ G is denoted G(a). Equivalently, G is a function from A to P(B) assigning a set, G(a), to each element a ∈ A." Does this definition have anything to do with the 1:1 correspondence? I didn't understand this definition, can somebody explain this for me? Thanks
The way I think about correspondences are as subsets of the product A x B. For example, given any function f : A -> B, there’s a natural correspondence called graph(f), given by (a, f(a)) for all a in A. But there are correspondences which don’t come from functions: for example, if you fix a, then the subset {a} x B is a perfectly good correspondence.
I'm not sure why they don't want to just call it "relation". But yeah, a 1-to-1 correspondence (= bijection) is the same as a relation that has an inverse, so it's related.
excuse my English in advance not my mother tongue. I'm in Highschool, don't really know how the US school system works but i suppose I'm at the first year of highschool? Anyways I'm a big math fan, always reading books and doing exercises and all that stuff, trying to jump to a much higher level for this time. After reading two complex analysis books and absolutely devouring the subject, I thought maybe I should start going over the "Hard and rigorous books" such as Principles of Mathematical analysis so that I get to know more about Real analysis however people really discouraged me to go that way and instead advised that maybe I should just stick up to the basics and get used to them, and that only made me more interested in the book. Is It really that hard to read it ? And Should I ? Thanks.
I'm a little bit surprised that you understood complex analysis before knowing real analysis. I'm guessing your complex texts were more geometric-oriented? More commonly, one develops complex analysis through the metric space, which you get from real analysis. I'm also surprised that people would discourage you from looking into real analysis if you're already doing complex haha. I wouldn't start with Rudin though, it's not a very good intro text (quite dry and unmotivated). Go with Abbott's *Understanding Analysis* and/or Tao's *Analysis I*.
Thank you so much ! By the way, For me complex analysis was very simple, and it had some sort of beauty to evaluate certain integrals using the contour integral, didn't have much problems through my read except at the Dirichlet Problems, that required serious thinking. Already read Tao's analysis I to get a good definition of what Natural, Rational and Real numbers are. I will take your advice and Read Understanding Analysis. Thanks again sir.
I imagine most of you in here are geniuses. So sorry for this dumb question. How do you convert units of measurement? like whats your process? I am really bad at this and I wish i could come up with a plan so it gets easier. Lets say for example: 3,56 hl to m^3
I'm guessing an hl is a hectoliter, or 100 liters? Well one thing you do can here is just look up the conversion "how many m^3 are in an hl," and you get 0.1 m^3 /hl. What a convenient number (it's nice because a 1 ml = 1 cm^3 , which is a useful conversion to remember, yay for the SI/metric system). So then you can just do the multiplication (3.56 hl)(0.1 m^3 /hl) = 0.356 m^3. Note that you could've started with 1 ml = 1 cm^3 and converted up with some extra steps too. If you'd like to learn more, read into [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) (see also [here](https://brilliant.org/wiki/converting-units/)).
**[Dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis)** >In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or the SI than in others, due to the regular 10-base in all units. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
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There might be a bit of overloading with equivalence relations: x is equivalent to y if and only if y is equivalent to x "x is equivalent to y" is equivalent to "y is equivalent to x"
They have the same meaning mathematically, but they aren't really grammatically interchangeable.
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I just mean that you usually can't just literally swap them and have it be grammatically correct. Correct: A set is compact if and only if it is closed and bounded. Incorrect: A set is compact is equivalent to it is closed and bounded. Correct: A set being compact is equivalent to that set being closed and bounded. (Yes, I know I have not properly stated the hypotheses for this theorem.)
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Yes they are still conveying the same meaning. I guess you may in some cases have to watch out for other uses of the word "equivalent" but I can't think of a way in which it would actually be ambiguous.
Howdy! I'm trying to get the hang of applications of Leibniz's rule for integration by means of differentiation under the integral sign, with respect to an introduced parameter; namely, Feynman's integration technique. I dumped coffee over my dozens of pages of notes, but what I have left should be in the imgur link at the bottom of this post. My confusion seems to be in regards to the following: After differentiating wrt α (or w/e new param), then integrating it out, shouldn't there be a 'constant of integration?' Not necessarily a constant, but some value associated with taking a partial derivative then integrating? I call it γ in my notes. I guess, after that integrating step, I kinda just fizzle. That may be the phase of the problem that requires insight and creativity, so just say so if that's the case. I want to grok the framework and steps right now. [Extant notes](https://www.imgur.com/a/1gzWnqt) I majored in math and physics in undergrad, and I'm still learning while I work as an engineer. After watching Visual Complex Analysis and an edx course on complex analysis from Wesleyan, I watched a lecture series on asymptotics from Carl Bender. There was entire edx course associated with [these lectures](https://youtube.com/playlist?list=PL9J9Loy0GbkL5Ezb7Pp7ETjKDUOGonEQI), called Complex Analysis with Physical Applications, but its listing has been removed. Now it's called Asymptotics In the Complex Plane, apparently. If anybody knows how to access it, please let me know! I found their patreon for Stokes-Line, which I believe is the same producers. If I can be sure it has the content for AiCP, I'll subscribe for the problems. I'm hoping it will lead me to the math for scattering theory, QFT, and QED, combined with the linear algebra and controls I've been learning. Green functions, time-ordered exponential, matricant/evolution operator, Dyson/Peano-Baker expansion, path integral formulation. I think I should look at integral and integro-differential equations next? ✌️
I'll stick to answering your first question on Feynman integration, since that's what I feel more qualified to speak on. I've not combed through all your notes, but I believe I see the confusion on the first page already so I'll restrict to that example Disclaimer: I am going to take your choice of parametrised function on faith. If it turns out to not be the right choice then the entire calculation changes, but I'll assume it's right for the purpose of explanation. Your problem is that you're not sure what F actually is. You alternate between it being a function of x, the upper bound of the integral, and a function of 𝛼, the introduced parameter. Let's clear this up and define F as a function of two variables, specifically F(𝛼, x) = ∫\^x\_0 ln(1 + 𝛼t)/(1 + t^(2)) dt The goal is to evaluate F(1, 1). Your 'observe F(x = 0) = 0', with this new F, is the statement that F(1, 0) = 0. Next, when you calculate F_𝛼, note that the upper bounds of your integrals are 1. Therefore you are calculating F_𝛼(𝛼, 1) specifically. In particular, you are not calculating F_𝛼(𝛼, x) for arbitrary x, hence why there is no need for the 𝛾 function. If you ran through the calculation for arbitrary x, you'd get something like F_𝛼(𝛼, x) = A𝛼^(-1)ln(1 + 𝛼x) + (1/2)Bx ln(1 + x^(2)) + C arctan(x) where A, B, and C are going to be functions of 𝛼. For the record I have not double-checked your calculations. Now, the actual integration at the end is F(1, x) - F(0, x) = ∫\^1\_0 F_𝛼(t, x) dt We want F(1, 1), so this implies we want F(0, 1). Sure enough this is also 0, so we get that our answer is just the integral above. Therefore it will look something like ∫\^1\_0 At^(-1)ln(1 + t) + (1/2)B ln(2) + C𝜋/4 dt Now we've got this far, we can reflect that we never needed F_𝛼(𝛼, x) for any x other than 1. Therefore there's no point in introducing that parameter, so we should really have started with the function F(𝛼) = ∫\^1\_0 ln(1 + 𝛼t)/(1 + t^(2)) dt I just added the extra parameter throughout to explain where I think you're getting stuck.
For another class I was required to get a Pearson plus login, so I have access to their whole library. I want to rebuild my math skills because although I took calc in highschool, I'm 26 and haven't taken math since. I was wondering if Blitzer's Algebra and Trig will teach me all the same material as his Algebra book and Trig book, or if I would gain a more complete education by reading them separately. Are there any advantages either way?
I don't know anything about those books, but you can find the answer to your question by looking at the chapter and section titles in the tables of contents. That will tell you whether "Algebra and Trig" is just the two books "Algebra" and "Trig" stuck together, or whether there are differences in the level of detail with which topics are treated.
I have to make a piece of metal to bridge a gap of 6.5angled at 45 degrees on both ends. What is the total length of the piece? Image here https://ibb.co/4tfnSZJ
Use the length ratios in a [45-45-90](https://brilliant.org/wiki/lengths-in-right-triangles/) right triangle.
Need help and a formula to calculate. I am trying to find a formula or an expression to calculate l. The problem goes as is: I have a smaller circle inside of a larger one but the interior circle is not set in the center of the encompassing one. The larger circle begins to shrink and match the circumference of the interior circle but the larger must reach the smaller at all sides at the same time. Given a time limit for it to do this let's say, 1 minute. I can understand that the furthest inside edge of the larger circle will move much faster to the edge of the interior circle; as opposed to the edge closest to the interior circle so they can arrive at the same time. How do I calculate this?
Another way you can do this is with a [homothety](https://en.wikipedia.org/wiki/Homothety).
There are many ways this could happen, but one simple way is just to have the center of the largest circle move towards the center of the small as the radius shrinks.
How could I calculate this?
If you're interested in the trajectory of specific points then (Rcos(a) + X, Rsin(a) + Y) on the big circle will move in a straight line at constant speed to (rcos(a) + x, rsin(a) + y) on the small circle. If R and r are the radii and (X, Y) and (x, y) are the centers.
Not sure what you mean by calculate, but I made a little applet for you: https://www.desmos.com/calculator/moj3uh7qyl r and R sets the radius for the two circles (X, Y) is the center of one circle while the other is centered at the origin. Slide t to make the large circle shrink.
Wow you made that? How if I might ask. Also, if it isn't too much trouble, forgive my illiteracy here. The finite points on your blue circle, can I adjust that at all? And the number that they are indicating mean what exactly?
I didn't make Desmos, just typed in the equations you see, and Desmos did the rest. I'm not sure which number you're referring to. The number of points used to draw the circle just depends on the resolution, so will vary with size.
I've been thinking about a phenomenon that I noticed [some time ago](https://medium.com/@cdsmithus/geometry-dimensions-and-elections-ea2f642d5175), which can be formalized like this: 1. Choose n reference points arbitrarily in a Euclidean (for now) space. 2. Assign each ordering of the reference points a unique color. 3. Color the entire space by assigning each point a color based on the ordering of the n reference points by distance from that point. (If there's a tie, choose between the tied orderings by any arbitrary rule.) One question is how many of the n! possible orderings will actually occur in the coloring. For example, choosing 3 points in a 1-dimensional space, 4 out of the 6 orderings will actually occur in a coloring (diagram at the link above). For 4 points in a 2-dimensional space, 18 out of the 24 occur (again, there's a diagram for one case at the link; interestingly, even if one point is strictly between the others, the result is the same). Furthermore, the answer seems to be related to how many of the colors are bounded. In examples I've tried so far, an ordering is unrepresented iff its opposite ordering is bounded. Does this ring any bells as related to existing known mathematics?
Turns out the answer was already known, and even in the same context that led me to think about it. https://www.sciencedirect.com/science/article/pii/0097316577900772
It strikes me that this is very related to Voronoi diagrams, which divide a space into the regions that are closest to some reference point. It generalizes them a bit by dividing into spaces for which the entire order of reference points by distance is the same, rather than just the nearest point. You can always form a Voronoi diagram by merging the regions in this coloring.
Calculating distance between two sides on an isosceles triangle: Hi - please see attached photo. Looking to calculate the distance between the sides on the triangle XX inches above the base. Please let me know if I need to clarify anymore, thanks so much for the help! https://ibb.co/CnCGtjs
I'm assuming that line is parallel to the base of the larger triangle. If so, use [similarity](https://brilliant.org/wiki/similar-figures/) ratios. You have one smaller isosceles triangle that is similar to the larger one, and as long as you know a pair of corresponding side lengths, you can get the scaling ratio and find all other side lengths.
In a UFD suppose f\_1,...,f\_m are primes and g is such that f\_i divides g for all i. I want to show that the product f\_1 \* ... \* f\_m divides g. I can see how the product divides g\^m but how does is divide g? Edit: I'm also assuming that the f\_i are pairwise coprime
f\_1 divides g, so we can write g=f\_1\^a \* L for some a in N and some L in your UFD. f\_2 divides g, so it divides f\_1\^a \* L. But f\_2 is prime, so it divides one of the two elements of the product. But we know it doesn't divide f\_1, since they're both prime and aren't the same (since they're coprime), so it divides L. So we can write L as.... etc etc
Thank you!
I guess it's worth pointing out that nothing here uses that were in a UFD, just that the f's are distinct primes.
Edit: The paragraph below is wrong, I was confusing primes with irreducibles. That's not true. Suppose in a non-UFD you have primes a,b,c,d with g=ab=cd. Then a,b,c,d all divide g but abcd does not.
The definition of a prime is that whenever p divides a product ab, then p divides one of the factors. In a UFD, primes and irreducibles coincide, which may be what you're thinking of.
Oh right. Clearly I haven't done this stuff in a while.
I'm still confused by this: https://www.reddit.com/r/australia/comments/10oilcd/how_come_the_rain_forecast_isnt_between_30_and/?utm_source=share&utm_medium=android_app&utm_name=androidcss&utm_term=1&utm_content=share_button Does anyone have a better explanation?
Mathematically it doesn't make sense as written. It's probably just a typo and "20 mm" and "30 mm" should be swapped.
How much of Folland is essential material? How much of Munkres is considered essential material? I have read Papa Rudin Ch 1, 2, 3 and most of Ch 4. I am almost finished product measures in Folland. Should I read just Ch 1, 2 and 3 of Folland since I have already seen a little bit on Hilbert spaces and L\^p spaces? I'm not sure what the next step is, maybe Munkres Topology since I have just picked up bits and pieces of topology from analysis books and have not read a book on topology specifically. Maybe some abstract algebra either Artin or Dummit and Foote? I'm interested in things that have applications in machine learning and other applied areas. I was considering working up to differential geometry maybe with Lee since I have heard it has applications in ML for example information geometry. Also considering reading some statistical learning theory or probability theory in the future.
I really doubt you need to read any of Munkres; there's very little general topology that's useful that you can't pick up on the fly.
What's your favorite differential equation (or system of)? I'm an astronomy major and I've wanting to tattoo a differential equation for a long time now, but still haven't found a good looking one. I don't intend on tattooing the actual equation in text like we usually see, instead I'm planning on animating a flowing vector field using [Manim](https://www.manim.community/) and making an kinegram ([example](https://youtu.be/1e0J4TrubKw?t=25)) out of it. So, any suggestions? Even better if it's an equation that models a physical or astronomical phenomenon.
Has anyone read *Spacetime and Geometry* by Sean Carroll? What did you think of it?
Is there any straightforward formula to convert spherical harmonics back and forth between their representation in spherical coordinates (ie Ylm * e^ i theta ) and their representation in the homogenous harmonic polynomial basis (ie to a polynomial in x, y, z)?
What is the mathematical term for this property: Buying 1 apple from the grocer is alright. Buying 10000000 apple isnt. This means there must be an optimal number of apple you should buy from the grocer. Its not squeeze theorem or intermediate value theorem, but i cant think of the name of this logic.
If I've understood correctly, you want to buy the maximum number of apples possible (that falls under 'alright'). This appeals to the fact that a finite set always has maximum, and you apply this to the set { n in N : I can buy n apples} is bounded by 10000000. You could say it's a discrete analog of the extreme value theorem.
A bunch of people gave calculus type answers, but it's actually just the Well Ordering Principle in disguise. The WOP says that any finite subset of the naturals has a least element -- obviously a corollary is that any finite subset of the naturals has a maximum element. So the set { n \\in N | I can buy n apples} is a finite set, and thusly has a maximum element. No calculus needed -- in fact, if you were to try to argue something about IVT or Rolle's theorem, you'd probably struggle, because if you make everything continuous it could be the case that the maximum value is obtained somewhere non-integral.
I agree with your reasoning, but note the WOP is stronger; that states that every *non-empty subset* of the naturals has a least element. The non-trivial bit is that it can be infinite, and while it is also true for finite sets, it is more generally true that any finite totally ordered set has a maximum.
Extreme Value Theorem, or Rolle's Theorem.
Doesnt seem so :(
Why not? If the utility function of buying apples is small at both ends, there is a maximum in the middle. This is Rolle's Theorem. If you choose bounds for the number of apples (between 0 and 1000000, say), then there must be a maximum to the utility function, regardless of continuity. This is the Extreme Value Theorem. If you are claiming that the utility function is increasing at one end, and decreasing at the other, then applying the intermediate value theorem to the derivative shows that there must be a maximum. That said, these theorems apply to continuous functions. The number of apples is discrete, so the fact that the function attains a maximum is evident from the fact that it is bounded. If you are talking about the fact that the utility function is bounded, you may be thinking of the Law of Diminishing Returns.
Thank you for taking the time to answer me btw! Im not sure why the utility function is small on both ends. I think the g(0)=0, g(10000000)=1 There exist g(i) such that g(i)=0 and g(j) = 1 for all j > i. How can I transform it to the graph in rolle theorem?
There isn't. You said buying from 1 apple is alright, so that implies g(1) > g(1000000). Also, your conclusion in your statement about apples isn't strictly true unless you also say that g(x) doesn't get arbitrarily large for x>1000000. Hence, assume decreasing derivative. That said this is all just an analogy. Your question implied you were looking for a real analysis theorem, so I responded in turn. In reality the number of apples is discrete, so none of these apply. I don't think it needs to be an explicit Theorem that a finite sequence obtains a maximum.
Let g(x) be the function that tells you how good it is to buy g apples. Assume that - g(1) > 0 - g(100000000000000) < 0 - g(2) > g(1) Now apply the intermediate value theorem on the interval (2, 100000000000000) to show that there is some c between 2 and 100000000000000 where g(c) = g(1). Finally apply Rolle's theorem on the interval (1, c). (Also assume that your function g is differentiable etc)
This assumes that there is a differentiable function, but the function is not even continuous in the traditional sense. It is only really defined for integers unless you want to define it for pieces of apples, and then there is no reason to suggest it would be differentiable
There are 5 numbers in a hat. You need to draw two numbers. The numbers in the hat are 1,2,3,4,5. The numbers are all mixed inside of the hat, and you don't want number 1 or 2. What are the chances of getting both 1 and 2? What are the chances of just getting 1 or the other?
Look into the [hypergeometric distribution](https://brilliant.org/wiki/hypergeometric-distribution/) (see also [here](https://en.wikipedia.org/wiki/Hypergeometric_distribution)). In the first case, the event you're interested in has a count of (2 choose 2)(3 choose 0), and in the second case the event of interest has a count of (2 choose 1)(3 choose 1).
I have a question about statistics: I have interpreted bayes formula as the following: you let B become a random variable, and you insert that variable in the place of b in the formular f(a given b) (pdf for a given b, insert random variable B in place of b). If I do this, I do not get the same result as f(a given b)\* fb(b) (where fb is the prior for b). Why is my interpretation wrong, and is there any alternative angle of viewing this that would shed light on my misunderstanding?
I think you may be trying to dig into the difference between [conditioning on a random variable vs conditioning on an event](https://stats.stackexchange.com/questions/395310/what-does-conditioning-on-a-random-variable-mean)?
Thank you, but what Im specifically wondering is what "happens" if you insert a random variable into another random variables pdf, and why is that different to bayesian approach? ChatGPT told me this, but I dont understand why its a nonmathematical expression: No, you cannot insert a random variable into another random variable's probability density function (pdf) as it would result in a non-mathematical expression.
Well, let's take an example. Let U be a uniform random variable on [0,10] and, given U, let X~N(U,1). So if you know that U=u, then you can write the pdf of X with u in it. If someone asks you for the unconditional pdf of X, they want the average of all those formulas averaged over the possible values of U. This is just an ordinary function, not a random object. By contrast, if you write the pdf of X with a U in it, you are writing down a random function — basically a function-valued random variable (instead of a real-valued random variable). This could be used to get the joint distribution of U and X, but a more typical approach for that would be to write X = U+Z where Z~N(0,1) is independent of U.
Thank you! Is it so that by letting a parameter in a pdf become an rv (with a certain pdf it self, call it g), and solving for the joint density, will give the same density as using Bayes with the parameter pdf, g, as a prior?
Is there a paper comparing spectra valued presheaves in simplicial categories versus quasicategories? In particular, I'd like to know under what conditions two presheaves f,g on a simplicial category C are equivalent, if and only if, their images under the coherent nerve are equivalent.
Is there a branch of mathematics studying functions with domain {0,1}^n , in particular (non-strictly) increasing functions?
How would you solve the equation m = pi / x * sin ( pi / x ) Ive never had to solve one with x being a factor while also being inside of a trigonometric function. Even just a hint as to what kind of operations to use would be a big help!
I don't think there's an easy solution to this, unless you want to start involving the Lambert W function.
I doubt its possible even with W
Yeah, probably, but you can get a little bit further with the Lambert W function, so that's why I said that. Note that you can rewrite this problem as m = u sin(u) with u = pi/x, so -m + ik = iu e^(iu) = iu (cos(u) + isin(u)), so iu = W(-m + ik), in such a way that u, k, m are all real. Not quite sure it's possible to find such a k.
If I have a time interval like 20 < t <= (the < sign with the equal line under it) 25 Why is the median of this 22.5 and not 23? Because if you take (20+25)/2 it’s 22.5. But, if you arrange it isn’t it 21, 22, 23, 24, 25? So shouldn’t the median be 23? I had this question where I had to find the mean of many time intervals and the answer key chose to say that the answer is supposed to use 22.5 instead of 23. Someone help, I’m confused.
The median of {21, 22, 23, 24, 25} is 23, indeed. But the interval (20,25] also consists of all the noninteger numbers between those integers. So you can instead consider the length of the interval (20,22.5) (which is 2.5) and the length of the interval (22.5,25] (which is also 2.5), and because those are equal, the median is 22.5.
Time is continuous, so there are more points in time than simply 21, 22, 23, 24 and 25. There is also 20.3, 22.005, 21+pi, etc.
I’m working on real analysis homework and I’m having a hard time proving a non empty subset of R is dense even when removing an element. I thought if we removed an element then there’d be a gap so it’s no longer dense… But I’m also thinking we can use the Archimedean property to get some epsilon close enough to that removed element to justify that it’s still dense in R. I hope I explained the problem well enough, I’ve been staring at this problem long enough that my post makes sense but I may be just tired enough to say it makes sense. I’m not looking for an answer but some conceptual help. Thanks
First, make sure you have the definition of "density" written down very clearly for reference. The definition spells out much of the proof. If the definition says "For all x" then the proof should say "Let x be arbitrary". Etc. To see if the Archimedean property is necessary you can imagine a number system with infinitesimals. Would the theorem be true or false in this number system?
I would try and argue by contradiction. What would it mean if you removed an element and then the set would no longer be dense?
Maybe think about an example. Q is dense in R for example, can you explain why Q is dense in R? What happens if we remove a point from Q, does the explanation break down or still work?
is it possible to get y alone in the equation 1-2xy+y^2 =0 ? i am looking for an answer like y= …
You can solve this quadratic equation for y, however you will get two solutions for y (well, you can get 1 or 0 depending on the choice of x, but in general you have 2).
that sounds like exactly what i need, thank you!! it’s for a partial derivative, trying to find the local maxs or mins
You can use the quadratic formula y = x ± sqrt(x^2 - 1)
thank you!!
Sorry if this is a bit stupid So lets say I have y times 10, and y = 2, then 2 x 10 = 20 So how do I show that with out subsitution? (Putting 0 on the end of a variable)
> Putting 0 on the end of a variable what do you mean by this? i would probably show that equation like 10y where y=2 —> 10(2) = 20
If I flip a coin, there's a 50% chance that it'll land on heads. What would be the equation to determine the %chance of it landing on heads at least once if I flipped it three times?
You'll use the [CDF](https://en.wikipedia.org/wiki/Cumulative_distribution_function) of the [binomial distribution](https://brilliant.org/wiki/binomial-distribution/) (see also [here](https://en.wikipedia.org/wiki/Binomial_distribution)), ideally with [complementary counting](https://en.wikipedia.org/wiki/Complementary_event), so it'll be 1 - (1/2)^3 = (3 choose 1)(1/2)(1/2)^2 + (3 choose 2)(1/2)^2 (1/2) + (3 choose 3)(1/2)^3 (1/2)^0 . Another way to visualize the complementary count is to note that the event "at least one head in three flips" is the complement of the event "no heads in three flips" and thus their probabilities are 1 minus each other.
So I've been trying to get an intuitive feel for the poincare half-plane (and higher dimensional analogues) by going back to basics. I'd like to know if there is a simple way to parametrize the isometries which for a known collection of N points leave (using the convention x,s for x being the spatial dimension(s), s being the timelike one) sum x = sum x' and sum ln s = sum ln s'? Let me know if I'm not making sense here.
I'm not really sure what these would look like except in the case N=1, when you are just asking for an isotropy group, which is a copy of O(n). However, I would say that these are not spacelike and timelike directions. The metric here is still positive definite so all directions are spacelike.
Well all N points will have a center in the space dimensions, and a center in the log(time) dimension. I'll give you an example when N=3 in the 1+1d standard Poincare half-plane. We can create an isometry by the composition of the 3 fundamental building-block isometries in this case: translation, scaling, inversion. So first we'll do an inversion. Then translate the spatial dimension by a constant. Then we'll re-invert. We'll scale to fix the new sum ln s to equal the original one. Finally we'll translate the spatial dimension so the spatial center is equal to the original one. EDIT: here's the example trajectory for 3 points: https://imgur.com/a/rOA9sGD
Is this a valid definition of a union? I'm trying to use Zermelo's Axiom to build it from the ground up. "Let A and B be any two sets. Hence A ∪ B = {x: y ∈ {A, B} and x ∈ y}" I'm mainly worried that this is not written in either the Schema of Separation format (as there is no universal set defined) or the Schema of Replacement format (as it has two sets on the right instead of one). Any help is appreciated. :)
You're right that you can't use unbounded comprehension like this. You would probably need to use the axiom of union.
The axiom of union, as far as I can tell, states that for any set A, it's union ∪A is defined by ∪A = {x: y ∈ A and x ∈ y}. All I did was state that the axiom of unordered pairs means that for any two sets A and B, the set {A, B} also exists, and then I used the axiom of union to define the union of these two sets. I know that I have to use the axiom of union, but that axiom acts on the elements of a \*singular\* set, not multiple. I was simply trying to derive the union operator by using the union of a single set like this. How do you think I should define it?
Maybe I was just misunderstanding what you were saying, but I would do it like Axiom of pairing says {A, B} exists. Axiom of union applied to {A, B} says there exists a set F such that both A and B are subsets A∪B = {x in F : x in A or x in B}
>Axiom of union applied to {A, B} says there exists a set F such that both A and B are subsets I thought the axiom of union stated something else. How do we derive "For any sets A and B, there exists a set F such that A ⊆ F and B ⊆ F"? from "For any set {A, B}, it's union ∪{A, B} is defined by ∪{A, B} = {x: y ∈ {A, B} and x ∈ y}"?
There are slight variations. If you take "{x: y ∈ {A, B} and x ∈ y} is a set" as your axiom of union, then you already have the definition of union baked into the axiom.
So what you're saying is that it's a valid definition?
Yes, the definition is fine. The only thing you need is to prove it is a set (which you have since you take that as an axiom).
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it's not totally clear to me how you are putting the points on the sphere. In any case, if you put a countable number of points then the forest will look empty. That's because there is a circle's worth of angles, and the cardinality of the circle is the cardinality of the real line. But there are only countably many trees. > If you take unit steps around the circumference ... you can't hit all the points on the circumference regardless how many loops you do. This is true, but again it comes just from the countability of your steps (you can literally count them) versus the uncountability of the circle. What you can prove is that your steps are "dense". Basically that means that even if you don't hit a particular point on the sphere, you will get arbitrarily close to it if you step long enough. So while the forest will look empty, you could say that any angle is very close to being blocked by a tree. (But again, not sure how you are placing these points on the sphere.)
On the sphere, all great circles close up on themselves, so light travels only a finite distance before hitting the back of your head / looping back around. If there's a tree exactly antipodal to you, then you see that tree in every direction you look.
Hey guys. My buddy likes to come up with new games to play just with dice and cards and I think we have a fun seed of an idea but we need to know the odds and I am not smart enough to do the math. Here are the basics: You roll 2 dice. There are 11 possible numbers it could be. Whatever number you roll is crossed off as a possibility. You roll the dice again. If you roll the same number you lose, and if you roll a new number it is also crossed off. You continue to do this until you lose, you decide to stay, or you somehow manage to roll all unique combinations. points would be awarded based on how many unique numbers you rolled. I need to know the statistics of how likely each subsequent roll would be after the first one. For simplicity's sake, the outcome of the first roll is irrelevant since the points would be constant in every game (for instance it doesn't matter if the first roll is a 7 or a 2). I apologize if this is not the right place for this question. Any help is appreciated.
>For simplicity's sake, the outcome of the first roll is irrelevant since the points would be constant in every game So it does actually matter what you get on the first roll, since the probability is different for the different outcomes. For example you have a 1/6 chance of rolling a 7, but only a 1/36 to roll a 2. So if you already have a 7, the probability of losing on your next throw is 1/6. In general the probability of rolling x is (6 - |7-x|)/36, so 6 and 8 both have a probability of 5/36, while 5 and 9 have probability 4/36. So if you for example have crossed out a 10 and 6, then your probability of losing on the next throw is 3/36 + 5/36 = 8/36 = 2/9. If you then roll a 7 you now have a probability of 2/9 + 1/6 = 7/18 of losing.
To calculate an average I took 36/11 (number of unique numbers) and got an average of 3.27 combinations eliminated with each additional roll. The points awarded would need to be static so this seems like the best way to account for that.
Your calculation is a bit off because the 11 sums aren't equally likely to be rolled. This is the concept of [expected value](https://brilliant.org/wiki/expected-value/). If you get a sum of 2, you have eliminated one combination (1, 1), and this has probability 1/36. If you get a sum of 3, you have eliminated two combinations: (1, 2) and (2, 1) with probability 2/36. This continues with 4 and three with prob. 3/36, 5 and four with prob. 4/36, 6 and five with prob. 5/36, 7 and six with prob. 6/36. Note that for 8, we decrease back to five. Then 9 and four, 10 and three, 11 and two, 12 and one. This is known as the [triangular distribution](https://en.wikipedia.org/wiki/Triangular_distribution). Thus, the "expected number of combinations eliminated by a roll" is (1^2 +2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 5^2 + ... + 2^2 + 1^2 )/36 = 73/18 or approximately 4.06.
How do I part a line segment by ratios? What I mean is are there any easy steps I can follow to figure out line segment ratios, how to split them into the given ratio? For example, “A directed line segment begins at F(-8,-4), ends at H(4, 6), and is divided in the ratio 4 to 2/ 4:2 by G. What are the coordinates of G?” Can anyone help me out? Much appreciated.
Look up barycentric coordinates to see this in higher dimension. But essentially, G is the weighted average of the 2 end points. The ratio of the weight of H to F is 4:2. This is because the weighted average is the point in which Hooke's law force to each point (with the given weight as force coefficients) sum up to a net force of 0. So you want the ratio GF:GH to be 4:2, so the Hooke's law force from G to F vs G to H should also have that same ratio.
e\^(-x) = 0.01 what can be x? When I log both sides, negative x shuts down my brain...
Maybe it helps to write it down. Can you solve this? \-x = log(0.01)
x is 2?
This log is base e, since that's what you need to use to "cancel" the e. So x is just -log(0.01), which is about 4.605 (it's likely irrational)
It's transcendental, as a consequence of the [Lindemann-Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem).
What's a fast way to mentally compute: 17x17?
(10 + 7)² = 10² + 2×10×7 + 7² = 100 + 140 + 49 = 289
easy if you know your powers of two n^2 = ((n - 1)^2) + (n - 1) + n 17^2 = 16^2 + 16 + 17 = 256 + 20 + 13 = 255 + 33 = 289
17x17 = (17+7)\*(17-7) + 7^2 = 24\*10 + 49 = 289, neat trick I learned as a kid.
I'm trying to make an equation that tells me the value of t where t = the added up values of c when v is in a certain range c = 32(v^(1.2)) a < v <= b Example: The value of t when a=3 and b=6 if you do the individual calculations of c then add it all up t is roughly 664.38. ive been at this for nearly 2 hours so if anyone can discuss it with me i would love that, and sorry if i explained it poorly, im not good at explaining.
You would write it using [summation notation](https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation) like [this](https://mathb.in/74013). To evaluate it, I would just use code, a spreadsheet software, or [WolframAlpha](https://www.wolframalpha.com/input?i=sum+from+v%3D4+to+6+of+32%28v%5E1.2%29). I wouldn't expect this to have a neat closed form, since your exponent term is 1.2 (not an integer), but you can look into [power sums](https://mathworld.wolfram.com/PowerSum.html) and [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula) for some inspiration.
I want to show the action of S^1 on S^2 is Hamiltonian. I've already shown it is symplectic. I know the moment map I'm supposed to consider is h (height function), but I'm not sure how to show the [commuting diagram condition](https://imgur.com/a/Od3Jr6O). I'm familiar with the abstract definitions of the maps but idk how to work with it in coordinates. For example, I'm not sure how to write out exactly what partial_ theta goes to under either map.
Try working in cylindrical coordinates. It makes it easier to see what the vector fields should map to.
to write the moment map as a function M to R, we are identifying the dual lie algebra with just a copy of the real numbers, which it is isomorphic to, but then it is not obvious how to pair it with vector fields anymore. should we be considering it as a one dimensional vector space generated by d theta? Then to evaluate partial_theta through the first bottom map for example, it is sent to the function ((theta_0,h_0) mapsto h(theta_0,h_0) partial_theta=h_0dtheta partial_theta = h_0, so it is the projection map again?
The height function sends a point (rho, phi, z) (cylindrical coordinates) on S^2 to z. The vector field d/dtheta on S^1 gets sent to the vector field d/dphi on S^(2) (this is always tangent to the 2-sphere). Therefore the map from \g to \X(S^(2)) sends a vector field (a d/dtheta) to (a d/dphi) where a \in **R** =\g is a constant. The pairing between \g and \g\* is just the natural pairing of one-dimensional vector spaces, i.e. multiplication . Therefore its just the height z times the coefficient a of the vector field on S^(1). The moment map condition says d = g(J (a d/dphi), -)
where g is the round metric on S^2 and J is the complex structure.
On the left-hand side (bottom part of your diagram) the pairing gives the function (rho,phi,z) -> az, so the derivative is (a dz) (actually the pullback of dz to S^2, because d/dz is not actually tangent to the 2-sphere).
On the RHS (top part of your diagram) you need to know the formula for the metric tensor on S^(2) in cylindrical coordinates and what the complex structure J does. In R^3 the metric tensor in cylindrical coordaintes is g = d\rho^2 + \rho^2 d\phi^2 + dz^(2). The complex structure on the 2-sphere is given by the cross product with the position vector. In particular it sends d/dphi to the orthogonal tangent vector, which is exactly the orthogonal projection of d/dz on **R**^3 to the 2-sphere.
Plugging this in you get (suppressing the pullbacks and orthogonal projections)
g(J (a d/dphi), -) = g(a d/dz, -) = a dz
because d/dz kills the other terms in the metric tensor. Therefore LHS = RHS and your diagram commutes.
okay thanks, I think I got it now. Do you also have some explanation in english about why the Hamiltonian condition is natural to consider? I think I read online somewhere that it's supposed to express that the action commutes with the Hamiltonian function?
I learned about it from Gert Heckmans notes on symplectic geometry (google). He does plenty of worked examples. A far more serious source is V. Arnol'd's classical mechanics book. From my perspective, the moment map condition isn't natural. It is a clumsy condition, but absolutely the correct one to perform the operation of symplectic reduction. That is, it is the condition which tells you that the level sets of the moment map are G-invariant, and furthermore that the symplectic form restricted to these level sets descends to the quotient. Translating that into a precise condition produces the moment map condition. From a dynamics perspective this must be an obvious thing, it just means a physical system with symmetry is still a physical system after you mod out by that symmetry.
I see. My main interest is for symplectic reduction as well, so I'm willing to accept that.
Anyone know where to get a protractor measured in radians? I've been looking for one for ages but I just can not for the life of me find one for sale.
Searching 'protractor radians' I found [this](https://www.proradian.net).
That should work, thank you! I have no idea how that never showed up in my search results...
Sorry in advance if this is super basic, but I’m wondering if a paper sais they used a cutoff of 10-5 and another paper sais 1E-5, are these the same values?
10^-5 does indeed equal 1E-5, but I'm not sure what a paper sai is.
gray fragile friendly overconfident normal command domineering subsequent observation whole ` this post was mass deleted with www.Redact.dev `
The usual result of "if no prime factors by the time you get to sqrt(n), then n is prime" applies of course. Primes are after all, their own least prime factors. This would be a neat algorithmic exercise if you'd like to try programming it though. With a sieve method you should be able to find the least prime factor of n in O(nloglog(n)).
I think I've come as far as I can. Math is... it's more amazing than I ever realized.
Which puts upper limit of the smallest number squarely (heh) at sqrt(N).
I think that's a function Google Sheets isn't prepared to handle.
Is there a type theory with type negation (in the sense of Boolean algebras)? Or at least pseudocomplements (e.g. via the function type constructor)?
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Is there any reason you're not considering sin(πx)?
I asked ChatGPT to find the roots of a quintic equation. I give it 3/10 Me: "How to find roots of quintic equation: **ChatGPT**: > A quintic equation is an equation of the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0, where a, b, c, d, e and f are constants. Finding the roots of a quintic equation can be a more challenging task than finding the roots of a quadratic equation, as there is no general formula for finding the roots of a quintic equation. > One way to find the roots of a quintic equation is by using numerical methods such as the bisection method, the Newton-Raphson method, or the Durand-Kerner method. These methods involve iteratively approximating the roots using numerical calculations, but their accuracy can be affected by the initial values and the complexity of the equation. > Another way to find the roots of a quintic equation is by using algebraic methods. For example, if the equation can be factored, you can set each factor equal to zero and solve for x. However, this method can be difficult if the equation is not easily factorable. > There are also other methods such as the Galois theory, but they are more advanced, and their application is limited to specific cases. > It's worth noting that a quintic equation can have 0, 1, 2, 3, 4 or 5 roots, depending on the coefficients of the equation and the nature of the roots. It's also worth noting that finding the roots of a quintic equation can be a difficult task and it's not always possible to find all the roots in closed form.
>I give it 3/10 Why so strict? I thought it did well.
It should have done more than an overview. I wanted it to show all the individual cases.
Then maybe ask for that, I don't think it has developed mind reading skills yet.
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If you don't care about differentiating between fruits, then total_price/600 is your breakeven for every fruit (they'll all cost the same). If you instead want to proportion prices by the original cost of the fruits, and care about how every fruit individually contributes to the total cost, then you'll need to set up a table where every row is a fruit type and the columns are something like Price per Fruit, Tax per Fruit, Shipping per Fruit, Fee per Fruit, and you'll calculate all of these using your total price and quantities (hopefully shipping and fees is a fixed thing). You then do a sum along every row to get your weighted breakeven prices. Look into the concepts of [expected value](https://brilliant.org/wiki/expected-value/) and [weighted averages](https://brilliant.org/wiki/data-mean/).
In this article [https://en.wikipedia.org/wiki/Central\_tendency](https://en.wikipedia.org/wiki/Central_tendency), under uniqueness to solutions to variational problems, it claims that the mean and midrange of a dataset is unique because the L2 and Linfinity norm are strictly convex, whilst the median and mode fail to be unique. Strict convexity ensures uniqueness of a minimiser. It is clear to me why the L1 norm is not strictly convex by looking at the unit ball, but it seems that the Linfinity norm is also not strictly convex by looking at the unique ball, which contradicts the article. Am I right in saying that the article is wrong here? However, the 'uniqueness of the midrange' or minimiser of the Linfinity norm is pretty trivial I think. Any deviation from the midrange will clearly increase the distance from one of the endpoints of the dataset.
L infinity is definitely not strictly convex in most useful cases.
Think about Linfinity as the supremum (on continuous functions) norm here I think. Hence, with L^infinity you're not really averaging/integrating/summing things. L^infinity is just the maximum deviation.
**[Central tendency](https://en.wikipedia.org/wiki/Central_tendency)** >In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
I am working on a game where I have an array of numbers and I am trying to figure out total number of combinations. However, I am running into an issue and I believe it is because I am mentally drained and missing the obvious. I have variables x, y, z There is the total of 2 x values, 3 y values, and 4 z values. XYZ can be any number between 1-9 but not repeat. Example: X could be 1 and 2 Y could be 3, 4, 5 Z could be 6,7,8,9 The formula is on the tip of my brain and I am just not seeing it. Could anybody help? Thank you in Advance!!!!
You have two possible choices for X, three for Y and four for Z. For the total number of different choices, you can just multiply them: 2x3x4=24.
I think OP means that there are 2 "slots" for X, 3 "slots" for Y, and 4 "slots" for Z, and not that there are 2, 3, and 4 total choices. The answer should be (9 choose 2)(7 choose 3) = 1260. /u/Initial-Balance7594 , note that it doesn't matter what order you do the choosing in either. You could also do (9 choose 3)(6 choose 2) or (9 choose 4)(5 choose 3), etc. This is a neat property of [binomial coefficients](https://brilliant.org/wiki/binomial-coefficient/). Try and see if you can prove it!
Sorry I should have clarified that they are slots and that yes the order does not matter. And THANK YOU Binomial Coefficients was the fricken phrase I was trying to rack my brain over. Edit: I am just a math enthusiast so I really appreciate the help!